Problem: What is the area of the region between the graphs of $f(x)=x+1$ and $g(x)=\dfrac{25}{x^2}+x$ from $x=1$ to $x=5$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $16$ (Choice B) B $25$ (Choice C) C $\dfrac{184}{15}$ (Choice D) D $\dfrac{392}{5}$
Answer: Visualizing the area We sketch the graphs of $f$ and $g$ first. ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${5}$ ${10}$ ${15}$ ${20}$ ${25}$ $f$ $g$ $y$ $x$ From the graph, it appears that $g(x)\ge f(x)$ between $x=1$ and $x=5$. From this we are looking to evaluate: $ \int_{1}^{5}\left( g(x)-f(x) \right)\,dx$ Evaluating the definite integral $\begin{aligned} &\phantom{=} \int_{1}^{5} \left( \dfrac{25}{x^2}+x - (x+1) \right) \,dx \\\\ &= \int_{1}^{5} \left( \dfrac{25}{x^2}-1 \right) \,dx\\\\ &= -\dfrac{25}{x}-x~\Bigg|_{1}^{5} \\\\ &= \left( -5-5 \right) -\left( -25-1 \right)\\\\ &= 16 \end{aligned}$ Answer The area is $16$ square units.